A domed ceiling is a parabolic surface. For the best lighting on the floor, a light source is to be placed at the focus of the surface. If 2 m down from the top of the dome the ceiling is 28 m wide, find the best location for the
Light source.
A) 49 m down from the top
B) 73.5 m down from the top
C) 98 m down from the top
D) 24.5 m down from the top

Question

Quizplus · Accepted Answer

The best location for the light source in a parabolic surface is at its focus. The equation of a parabola in vertex form is $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex of the parabola. For a parabolic dome, we can simplify this to $y = ax^2$ if we assume the vertex is at the origin and the parabola opens downwards. The focus of a parabola is located at $(0, \frac{1}{4a})$ from the vertex. Given that the dome is 28 m wide (14 m radius) 2 m down from the top, we can use the equation $2 = a(14)^2$ to find $a$. Solving for $a$, we get $a = \frac{1}{98}$. The focus is at $\frac{1}{4a} = \frac{1}{4 \cdot \frac{1}{98}} = 24.5$ m down from the top, which is where the light source should be placed for optimal lighting.

A Domed Ceiling Is a Parabolic Surface